Mr Grace, A Theorem on Curves in a Linear Complex. 133 



For n > 5 a different method is necessary, and it applies 

 equally well to n = 3, 4, or 5. 

 Suppose the curve is denned by 



#=/iO)> y =/»(*■)» *=*'/*($), w =/«(*), 



where A, is a variable parameter and the /'s are rational integral 

 functions of order n. 



Then the coordinates of the tangent at any point of the curve 

 are the six Jacobians of they's taken in pairs, i.e. 



"23) "31) "12) "14) "24> "345 



and each of these is of order 2 (n — 1), so that the rank of the 

 curve is 2 (n — 1). 



Now consider the points of the curve at which a linear com- 

 plex contains six consecutive tangents ; it is easy to see that their 

 parameters are given by 



"23) "31) "12) "14) "24) "3, 



a J 23 



~dX 



d s J K 



= 



(A), 



d\ 3 



a J 2z 



~dxF 



~dtf 



and that the order of this equation in X is 6 (m — 5) where m is 

 the order of each J. (Of course the easiest way to see the last 

 fact is to make the J's homogeneous for a moment by the addition 

 of a variable /x.) Since m = Zn — 2 we infer that there are in 

 general 12w— 42 such points on a rational curve of orders. If 

 there are more than this number the equation (A) is an identity 

 and then all the tangents belong to a linear complex. 



Now in our case there are 2n — 6 stationary tangents and each 

 of these 1 counts for six of the points defined by (A); hence the 

 equation (A) has 6 (2?i — 6) = 12n — 36 roots and therefore it is an 

 identity. Consequently when there are 2?i — 6 stationary tangents 

 all the tangents belong to the same linear complex. 



1 Camb. Phil. Soc. Proc. xi. 28. 



